Question

in the figure above, chord ab is parallel to chord cd, the measure of arc ac is 48°, and the measure of arc dc is 138°. what is the measure of arc ab?
a. 126°
b. 48°
c. 138°
d. 122°

Explanation:

Step1: Recall circle - arc property

The sum of the measures of the arcs of a circle is 360°.

Step2: Set up an equation

Let the measure of arc \(AB\) be \(x\). We know that the measure of arc \(AC = 48^{\circ}\) and the measure of arc \(DC=138^{\circ}\). Since \(AB\parallel CD\), we use the fact that the sum of the arcs \(AC + AB+DC\) and the remaining arc (which is part of the circle) gives 360°. In this case, we can also use the property of parallel chords in a circle. The sum of the non - overlapping arcs between the parallel chords and the arcs formed by them is related. The sum of the measures of the arcs of the circle implies \(x + 48^{\circ}+138^{\circ}=360^{\circ}\).

Step3: Solve for \(x\)

\[

$$\begin{align*} x&=360^{\circ}-(48^{\circ} + 138^{\circ})\\ x&=360^{\circ}-186^{\circ}\\ x& = 126^{\circ} \end{align*}$$

\]

Answer:

A. \(126^{\circ}\)